# Determine the truth values of the following statements

• Jan 21st 2010, 08:43 AM
Runty
Determine the truth values of the following statements
Let S(a,b) be the statement "ab=a/b".

Given that the domain (or universe of discourse) for both variables consists of all positive, non-zero integers, what are the truth values of these two statements?

a) ∀a∃b S(a,b)
b) ∀b∃a S(a,b)

• Jan 21st 2010, 10:09 AM
hatsoff
Quote:

Originally Posted by Runty
Let S(a,b) be the statement "ab=a/b".

Given that the domain (or universe of discourse) for both variables consists of all positive, non-zero integers, what are the truth values of these two statements?

a) ∀a∃b S(a,b)
b) ∀b∃a S(a,b)

Well, we can put them into plain English:

(a) For every integer $\displaystyle a\geq 1$ there is an integer $\displaystyle b\geq 1$ with $\displaystyle ab=\frac{a}{b}$.

This is clearly true by choosing $\displaystyle b=1$ for any $\displaystyle a$.

(b) For every integer $\displaystyle b\geq 1$, there is an integer $\displaystyle a\geq 1$ with $\displaystyle ab=\frac{a}{b}$.

Now, regardless of $\displaystyle a$, it is clear that $\displaystyle b^2=1$. So if $\displaystyle b=2$ then (b) implies $\displaystyle 4=1$, which is a contradiction. So (b) is false.
• Jan 21st 2010, 11:33 AM
Runty
Quote:

Originally Posted by hatsoff
Well, we can put them into plain English:

(a) For every integer $\displaystyle a\geq 1$ there is an integer $\displaystyle b\geq 1$ with $\displaystyle ab=\frac{a}{b}$.

This is clearly true by choosing $\displaystyle b=1$ for any $\displaystyle a$.

(b) For every integer $\displaystyle b\geq 1$, there is an integer $\displaystyle a\geq 1$ with $\displaystyle ab=\frac{a}{b}$.

Now, regardless of $\displaystyle a$, it is clear that $\displaystyle b^2=1$. So if $\displaystyle b=2$ then (b) implies $\displaystyle 4=1$, which is a contradiction. So (b) is false.

Thank you very much. Glad to get that one out of the way.